On - line Complexity of Monotone Set

On-line models assume a player, A (randomized or deterministic), who makes immediate responses to incomming elements of an input sequence s = a 1 : : :a r. In this paper we study the case, when the response for every a i is a single bit, which is interpreted as pick/not-to-pick. The player's objective is to pick as many elements as possible under the condition, that the set of picked elements, A(s), satisses a certain prescribed property. Bartal et al. in 3] give bounds for the randomized player's worst case performance, when the incomming elements are vertices of a xed graph G, and the property in question is some hereditary graph property. In their model G is known to the player, but she does not know when the input sequence ends. In this paper we allow A(s) to belong to an arbitrary, but xed monotone set system M, where M is known to the player. We assign the performance measure e(M) = max A min s E(jA(s)j)=OPT(s), to every monotone set system M, where E() is the expectation over the player's ramdom choices and OPT(s) = max M2M jM \ sj. The above performance measure corresponds to the reciprocal of the competitive ratio. The question we raise in this article is a general one: what set theoretical properties of M determine the value of e(M)? It easily follows from earlier works, that e(M) = 1 ii M is a matroid, but in general it seems very hard to give a combinatorial characterization of set systems with close to optimal performance. The video on demand problem studied in 1] and 2], corresponds to the case, when M is the monotone closure of l disjoint sets of size k. Denoting the performance of this set system with e(k; l) we show that e(k; 2) = k=(2k ? 1), and that k=(3k ? 0:5 ? min z=1;:::;k (z=2 + k=z)) e(k; 3) k=(3k ? min z=1;:::;k (z + k=z)). Our extensive computer experiments suggest that the lower bound on e(k; 3) is sharp. Awerbuch et al. show that e(k; l) = (1= logk logl) and prove that this bound is optimal for a wide range of k; l pairs. We show 1 that e(k; l) k l?1 =(k l ? (k ? 1) l) which is bigger than (1= logk log l) for some values of k and l. One motivation for obtaining …